find an ordered pair (x, y) that is a solution to the equation. -x+6y=7

Answer:

Sarah earns R6 100 in total for her work to the value of R12 000.

Step-by-step explanation:

To calculate Sarah's earnings, we need to break down her income into two parts: the commission she earns on income up to R5 000, and the bonus commission she earns on income above R5 000.

Commission on income up to R5 000:

Sarah's basic salary is R3 000 per month, and she earns 20% commission on income up to R5 000. So for the first R5 000 of income, Sarah's commission is:

[tex]\text{Commission on income up to} \ R5, 000 = 20\% \ \text{of} \ R5, 000 = R1 ,000[/tex]

Bonus commission on income above R5 000:

Sarah also receives a 10% bonus on top of the normal commission rate on earning above R5 000. So for the amount earned above R5 000, her commission is:

[tex]\text{Commission on income above} \ R5, 000 = (20\% + 10\%) of (R12, 000 - R5, 000) = 30\% \ \text{of} \ R7, 000 = R2 ,100[/tex]

Total earnings:

Sarah's total earnings are the sum of her basic salary and the commission she earns:

Total earnings = Basic salary + Commission on income up to R5 000 + Commission on income above R5 000

[tex]\text{Total earnings} = R3, 000 + R1 ,000 + R2 ,100[/tex]

[tex]\text{Total earnings} = 6,100[/tex]

Therefore, Sarah earns R6 100 in total for her work to the value of R12 000.

Your total grade point average (GPA) for the semester is 2.67.

GPA stands for Grade Point Average. It is a numerical calculation used to measure the academic success of a student. It is calculated by taking the average of all grades received by a student across all courses taken in a given semester or academic year. Each course is assigned a specific number of credit hours, and each grade is assigned a numerical value based on the school’s grading scale. The numerical value of each grade is then multiplied by the number of credit hours for the course, and the total of all courses is added together to determine a student’s GPA.

A higher GPA is generally indicative of higher academic performance while a lower GPA is generally indicative of lower academic performance. A student’s GPA is used by schools, employers, and other organizations to evaluate a student’s academic record.

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This would give you a total of 29.0. Divide this by 10 credit hours and you get a GPA of 2.85 for the semester.

GPA stands for Grade Point Average. It is an academic measure of a student's performance in a course or program of study. It is calculated by dividing the total number of grade points earned by the total number of credit hours taken. A grade point average is typically expressed as a number on a 4.0 scale. A 4.0 GPA is considered to be the highest possible grade point average, while anything below a 2.0 GPA is usually considered to be failing.

This is calculated by adding up the total number of credit hours (10 credit hours) and then multiplying each grade by the respective number of credit hours. A=4.0, B=3.0, C=2.0.
So, you would multiply 4.0 by 3 for FYE 105, 3.0 by 3 for ENG 101, 2.0 by 3 for MAT 150, 2.0 by 3 for BIO 112, and 4.0 by 1 for BIO 113.
This would give you a total of 29.0.
Divide this by 10 credit hours and you get a GPA of 2.85 for the semester.

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Answer:

m₆ = 9375

Step-by-step explanation:

Given sequence is

[tex]m,_i = -3(-5)^{i - 1}[/tex]

To find the 6th term, all you have to do is substitute i = 6 and compute

For i = 6 we get

[tex]m_6 = -3(-5)^{6 - 1}\\= -3(-5)^5\\\\= -3(-3125) \\\\= 9375[/tex]

A negative number raised to an odd number is negative that is why (-5)⁵ is negative

Answer:

f'(x) = -6x^2

f'(-5) = -150

f'(0) = 0

f'(√17) = -102

The most likely to produce invalid statistical analysis of numeric measures is Pain rating as: none = 0; slight = 1; much = 2,  as it is an ordinal scale of measurement, which does not have equal intervals between the categories. So, the correct answer is B).

It means that the differences between the categories are not necessarily equivalent, and therefore, any statistical analysis based on this scale may not accurately reflect the true relationship between variables.

The other options (blood pressure in mmHg, oxygen saturation in percentage, neonatal birth weight in kilograms) are measured on interval or ratio scales, which have equal intervals between values and can be used for meaningful statistical analysis. so, the correct option is B).

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____The given question is incomplete, the complete question is given below:

Which of the following numeric measures would be most likely to produce invalid statistical analysis?A)Analysis of patients' blood pressures in mmHgB)Pain rating as: none = 0; slight = 1; much = 2C)Assessment of oxygen saturation in percentageD)Analysis of neonatal birthweight in kilogr

Answer:

D

Step-by-step explanation:

7/1 times 3/4 equals 21/4 cups of milk

Using probability, we can find that:

E(Y)= 2.4, Var (Y) = 0.64

E(Y) = 480, Var(Y) = 25,600

The probability of an event is the proportion of favourable outcomes to all other potential outcomes. To determine how likely an event is, use the following formula:

Probability (Event) = Positive Results/Total Results = x/n

Given,

The weekly CPU time is as follows:

f(y) where, 0≤y≤4

Here, probability density function is 4-y is correct, or else we get negative expected values.

We have to find E(Y) and var(Y)

E(Y) = 2.4

var (Y) = E(Y²)-(E(Y)) ²

= 6.4 - (2.4) ²

= 0.64

The CPU time is costing the firm $200 per hour.

Now, we find E(Y) and var(Y) of the weekly cost for the CPU time.

Y = 200E

E(Y) = 200 × 2.4

= 480

var(Y) = 200V(Y)

= 200 × 0.64

= 25600

We can observe that the weekly cost is not exceeding $600 as weekly cost for CPU time = 480.

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Answer:

pretty-pretty sure its 1/10

Step-by-step explanation:

Convert to a mixed number by placing the numbers to the right of the decimal over

10

. Reduce the fraction.

1

10

No element in this sequence can be divided by 5, 13, or 65.

This sequence's elements are not all divisible by 5, 13, or 65.

For positive integers n, we define the sequence a1 = 2n - 3.

We must determine if 2n - 3 is divisible by 5 for various values of n in order to determine whether members of this sequence are divisible by 5.

2ⁿ mod 5 equals 2ⁿ mod 1 = 2ⁿ mod 2 = 4ⁿ mod 3 = 3ⁿ mod 4 = 1, etc.

None of the items in this sequence can be divided by 5, as we can see from the fact that 2ⁿ mod 5 is not necessarily 0.

When divided by 13, 2ⁿ mod 13 equals 2ⁿ mod 1 mod 13 = 2, 2n mod 2 mod 4 mod 8 mod 3 mod 13 = 3, etc.

Since 2ⁿ mod 13 is not necessarily 0, none of the sequence's elements are divisible by 13 as a result.

When 65 is divided by 5*13, 2n mod 65 equals 2n mod 65 times 2, 2ⁿ

mod 65 times 4, 2ⁿ mod 65 times 8, 2ⁿ mod 65 times 3, etc.

None of the items in this sequence are divisible by 65 since 2ⁿ mod 65 is not necessarily 0.

Hence, No element in this sequence can be divided by 5, 13, or 65.

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The complete question is:

Consider the sequence a₁ = = 2¹-3=-1₁ -2²-3=1, 0₂= 03 =2³-3=5, 04-2¹-3=13, ⠀ a₁ = 2" - 3, defined for positive integers n. Which elements of this sequence are divisible by 5?

What about 13? Are any elements of this sequence divisible by 65= 5. 13? Why or why not?​

[tex] \boxed{ \red{ \mathfrak{A.\: 6 \frac{1}{6} }}}[/tex]

The scale of the map is 1 inch to 50 miles, or 1:50.

What is distance?

Distance is defined as the space between two points in space.

To find the scale of the map in inches per mile, we can use the ratio of the distance on the map to the actual distance:

5 inches on the map / 250 miles in reality

Simplifying this ratio gives:

1 inch on the map / 50 miles in reality

Therefore, the scale of the map is 1 inch to 50 miles, or 1:50.

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The formula that describes the vector field in which, x-axis is horizontal and the y-axis is vertical, is option B: F(x,y) = (y,1).

A vector field is a function that assigns a vector to each point in space. It can be represented as a group of arrows, each connected to a location in space and having a specific magnitude and direction.

The vector field that describes the given conditions is F(x,y) = (y,1). This means that at any point (x,y), the vector field will have a magnitude of 1 and will be pointing in the positive y direction.

The horizontal component of a vector is found by multiplying its magnitude by the cosine of the angle it makes with the positive x-axis. The vertical component of a vector is found by multiplying its magnitude by the sine of the angle it makes with the positive x-axis.

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Complete question is:

Which formula describes the following vector field? here the x-axis is horizontal and the y-axis is vertical. (Refer the image)

O F(x,y) = (1,7)

O F(x,y) = (y,1)

O F(x,y) = (1,x)

O F(x,y) = (x,1)

a) The sum of x and 4 equals 7: x + 4 = 7

b) The difference of y and 5 equals 4: y - 5 = 4

c) Two times x equals 10: 2x = 10

d) Two times y added to 3 equals 9: 2y + 3 = 9

f) If y > 3 + 2 or if 3 = y - 2 then y is smaller than y by 2

An equation is a statement in mathematics that two expressions are equivalent. It has two sides that are divided by the equals sign (=). One or more terms, such as integers, variables, constants, and mathematical operations like addition, subtraction, multiplication, division, and exponentiation, may be included on each side of the equation.

a) The sum of x and 4 equals 7:

x + 4 = 7

b) The difference of y and 5 equals 4:

y - 5 = 4

c) Two times x equals 10:

2x = 10

d) Two times y added to 3 equals 9:

2y + 3 = 9

e) If x exceeds 2 by 1, then either x = 3 or x > 2 + 1.

f) If 3 is smaller than y by 2 then either y > 3 + 2 or 3 = y - 2

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Answer:

5/12, 5/9, 2/3, 5/6, 6/5

Step-by-step explanation:

5/12= 0.41666667, 5/9= 0.55555556, 2/3= 0.66666667, 5/6= 0.83333333, 6/5= 1.2

Answer:

Step-by-step explanation:

We are given that the temperature readings are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. Let X be the temperature reading of a single thermometer selected at random. Then, X ~ N(0, 1).

We need to find the probability of obtaining a reading less than -2.74°C, which can be expressed mathematically as P(X < -2.74).

Using standard normal distribution tables or a calculator, we can find that the z-score corresponding to -2.74°C is:

z = (x - μ) / σ = (-2.74 - 0) / 1 = -2.74

The probability can be calculated as:

P(X < -2.74) = P(Z < -2.74) ≈ 0.0030 (rounded to 4 decimal places)

Therefore, the probability of obtaining a reading less than -2.74°C is approximately 0.0030.

Answer:

We need to find the probability of obtaining a reading less than -2.74°C from a normal distribution with a mean of 0°C and a standard deviation of 1.00°C.

Using the standard normal distribution, we have:

z = (x - μ) / σ

where:

x = -2.74°C (the reading we want)

μ = 0°C (the mean)

σ = 1.00°C (the standard deviation)

Substituting the values, we get:

z = (-2.74 - 0) / 1.00 = -2.74

Using a standard normal distribution table or calculator, we find that the probability of obtaining a z-score less than -2.74 is approximately 0.0030.

Therefore, the probability of obtaining a reading less than -2.74°C from the batch of thermometers is approximately 0.0030.

As a result, both varieties of trees will be the same height after 9 years. We can change both equations to a = 9 to verify this: Height of the type A tree is 36 + 8(9) or 36 + 72 inches, whereas the height of the type B tree is 18 + 10(9) or 18 + 90 inches.

Trigonometry includes heights and distances, and it has numerous uses in practical daily life. It utilised to compute height of towers, structures, mountains, etc., and distance between any two objects such celestial bodies others. ,sys,s tos.as to .... and.

Let's use "a" to denote the number of years after planting the trees.

A type A tree will reach the following height after "a" years:

Height of type A tree = 36 + 8a

After "a" years, the height of a type B tree will be:

Height of type B tree = 18 + 10a

We must set the two types of trees' heights equal to one another and solve for "a" to determine when they will reach the same height:

36 + 8a = 18 + 10a

Subtracting 8a from both sides, we get:

36 = 18 + 2a

Subtracting 18 from both sides, we get:

18 = 2a

Dividing both sides by 2, we get:

a = 9

Therefore, after 9 years, both types of trees will be the same height.

To check this, we can substitute a = 9 into both equations:

Height of type A tree = 36 + 8(9) = 36 + 72 = 108 inches

Height of type B tree = 18 + 10(9) = 18 + 90 = 108 inches

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Question:

You have two types of trees to plant in your yard: type A trees are 36 inches tall and grow 8 inches per year, while type B trees are 18 inches tall and grow 10 inches per year. At what point in time will the trees be the same height? How tall will the trees be at that time?

The number of functions ,

(a) that are one-to-one are 0.

(b) that assign 0 to both 1 and n are 2ⁿ⁻²,

(c) that assign "1" to exactly one of positive-integers less than n are 2.(n-1).

Part(a) : We have to find total number of "one-to-one" functions from the set {1,2,......,n} to {0,1}.

⇒ If n=1 then there are 2 possible functions depending whether 1 is mapped to "0" or "1" ,So there are 2 such functions.

⇒ If n=2 then domain is {1,2} then there are 2 choices for first element in domain.

Then, since one choice is taken there is one choice for second element in the domain. So, if n=2 we have 2×1 = 2 functions.

⇒ If value of n is greater than 2 then domain will be {1,2,....n} then only two value of this domain will be mapped to codomain {0,1} to provide a one-to-one function and

So, domain will not be used fully so there does not exist any one-to-one function.

Part(b) : Every element in the domain {1, 2, . . . , n} has two options in  codomain {0, 1},

So, there are total of "2n" functions from domain to co-domain.

Since, the function assigns 0 to both 1 and n.

There are "n-2" elements left in domain which can be assigned 0 or 1.

So, for "n-2" elements in domain and there are "2ⁿ⁻²" functions from domain to codomain.

Part(c) : The domain set has "n" elements and codomain set has "2" elements.

So, each of "n" elements from domain has 2 choices in function and thus we get "2n" total functions.

There are "n-1" elements less than "n" in domain.

Now, by the condition that exactly "1" positive-integer less than "n" maps to "1".

So, all other remaining less than n (i.e. n-2) must be map to 0.

We find this number in ⁿ⁻²C₁ ways = n-2;

So, total number of ways in which elements less than "n" can be mapped is = n-2(mapped to 0) +1(mapped to 1) = n-1

Also, "n" can be mapped to either "0" or "1" which means., nth element have two-choices.

So, there are 2.(n-1) possible functions.

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a) The moment generating function of X is

M(t) =

{ (1/(2-t)) e^(-2t) + (1/(2-t)) (1/(1+t)) if t<1

{ infinity if t>=1

b) The first moment (mean) of X is 3/4.

The second moment (expected value of X^2) of X is 7/8

(a) The moment generating function (MGF) of a random variable X with probability density function f(x) is defined as M(t) = E(e^(tX)), where E(.) denotes the expected value operator. Therefore, the MGF of X is

M(t) = E(e^(tX)) = ∫[0,∞) e^(tx) f(x) dx

Substituting the given probability density function f(x), we get

M(t) = ∫[0,∞) e^(tx) (e^(-2x) + (e^-x)/2) dx

Simplifying and integrating by parts, we get

M(t) = [(1/(2-t)) e^(-2t) + (1/(2-t)) (1/(1+t))] for t<1, and

M(t) = infinity for t>=1

Therefore, the MGF of X is

M(t) =

{ (1/(2-t)) e^(-2t) + (1/(2-t)) (1/(1+t)) if t<1

{ infinity if t>=1

(b) To compute the first moment (i.e., the mean or expected value) of X, we take the first derivative of the MGF at t=0

E(X) = M'(0) = d(M(t))/dt | t=0

Differentiating the MGF and simplifying, we get

E(X) = 3/4

To compute the second moment (i.e., the expected value of X^2), we take the second derivative of the MGF at t=0

E(X^2) = M''(0) = d^2(M(t))/dt^2 | t=0

Differentiating the MGF again and simplifying, we get

E(X^2) = 7/8

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The given question is incomplete, the complete question is:

Let x be a random variable whose probability density function is given by f(x) = e^(-2x) + (e^-x)/2 when x>0 f(x) = 0 when else, a) write down the moment generating function X (b) Compute the first and second moments

The final value of the integral as   [tex]$\frac{191}{60}$[/tex]   which represents the total "volume" of [tex]$f(x,y)=7x+6y$[/tex] over the region[tex]$D$.[/tex]

The given double integral can be rewritten as:

[tex]\iint_D 7x \, dA + \iint_D 6y \, dA[/tex]

where  [tex]D$ is the region bounded by $y=\sqrt{x}$ and $y=x^2$.[/tex]

To evaluate this integral, we first check the intersection points of the two curves:

[tex]\sqrt{x} = x^2 \quad \Right arrow \quad x = 0 \text{ or } x =1[/tex]

Therefore, the region  D is shown in the figure below:

region D

Next, we can choose to integrate with respect to [tex]x$ or $y$[/tex]. Since the lower boundary of  D is given by [tex]$y=\sqrt{x}$[/tex]which is a function of [tex]x$,[/tex] we choose to integrate with respect to x first:

[tex]\begin{aligned} \iint_D 7x \, dA &= \int_0^1\int_{\sqrt{x}}^{x^2} 7x \, dydx \\ &= \int_0^1 7x(x^2-\sqrt{x}) \, dx \\ &= \frac{56}{15} \end{aligned}[/tex]

Similarly, we can evaluate the other integral:

[tex]\begin{aligned} \iint_D 6y \, dA &= \int_0^1\int_{\sqrt{x}}^{x^2} 6y \, dydx \\ &= \int_0^1 6\left(\frac{x^4-\sqrt{x}}{2}\right) \, dx \\ &= \frac{49}{20} \end{aligned}[/tex]

Therefore, the value of the given double integral is:

[tex]\iint_D (7x+6y) \, dA = \frac{56}{15}+\frac{49}{20} = \frac{191}{60}[/tex]

In summary, we evaluated the given double integral by splitting it into two integrals, integrating with respect to x  first and then with respect to y , over the region bounded by [tex]y=\sqrt{x}$ and $y=x^2$[/tex]. By doing so, we obtained the final value of the integral as [tex]\frac{191}{60}$,[/tex] which represents the total "volume" of [tex]$f(x,y)=7x+6y$[/tex] over the region D.

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Answer:

(1,2)

Step-by-step explanation:

We are given the following system of equations:

y = -x + 3

y = 4x - 2

And we want to solve it, with the answer in an ordered pair.

Because both systems are equal to y, we can set both of the equations equal to each other, and solve for x in that way.

This is possible due to transitivity, which states that if a=b, and b=c, then a=c.

Hence,

-x + 3 = 4x - 2 (same as y=y)

We can add x to both sides.

3 = 5x - 2

Add 2 to both sides.

5 = 5x

Divide both sides by 5.

1 = x

Now, we can use this value to find y.

Substitute 1 as x in either y = -x + 3 or y = 4x - 2

Taking y = -x + 3 for instance:

y = -1 + 3 = 2

So, we now that x=1, y=2.

As an ordered pair, that is (1,2).

[tex]\text{order does not matter}[/tex]

[tex]\text{sample space}= \text{15 letters}[/tex]

[tex]\text{no repetition}[/tex]

[tex]\text{P(A)}= \text{15C5}= \text{3003 ways}[/tex]

Answer:

We can write the diameter and circumferance of base as -

D = 2√(750ρ/πh)

C = 2π√(750ρ/πh)

Step-by-step explanation:

What is function?

A function is a relation between a dependent and independent variable.

Mathematically, we can write → y = f(x) = ax + b.

Given is to find the diameter and height of the tin can.

Assume the density of coffee as {ρ}. We can write the volume of the tin can as -

Volume = mass x density

Volume = 750ρ

We can write -

πr²h = 750ρ

r = √(750ρ/πh)

D = 2r

D = 2√(750ρ/πh)

Now, we can write the circumferance as -

C = 2πr

C = 2π√(750ρ/πh)

Therefore, we can write the diameter and circumferance of base as -

D = 2√(750ρ/πh)

C = 2π√(750ρ/πh)

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When a shape is dilated, the size of the shape changes. The true statement is (d) The scale factor is 2.5.

Dilation:

Dilation is the process of changing the size of an object or shape by reducing or increasing its size by a specific scale factor. For example, a circle with a radius of 10 units shrinks to a circle with a radius of 5 units. Applications of this method are in photography, arts and crafts, sign making and more.

According to the Question:

How to determine the scale factor

In figure A, we have:

Length = 0.6

In figure B, we have:

Length =1.5

The scale factor is then calculated as:

K = 1.5/0.6

Dividing the equation:

k = 2.5

Hence, the true statement is (d) The scale factor is 2.5.

Complete Question:

The first figure is dilated to form the second figure. Which statement is true?

The scale factor is 0.4.

The scale factor is 0.9.

The scale factor is 2.1.

The scale factor is 2.5.

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On the 7th day after the videο was pοsted, 640 peοple will see the videο.

Statistics is the discipline that cοncerns the cοllectiοn, οrganizatiοn, analysis, interpretatiοn, and presentatiοn οf data.

1 Let's start by finding the pattern in the number οf viewers. We knοw that οn the first day, 5 peοple watched the videο. On the next day, the number οf viewers dοubled tο 5 x 2 = 10. On the third day, the number οf viewers dοubled again tο 10 x 2 = 20. We can see that the number οf viewers is dοubling each day, which means we can write the number οf viewers as:

[tex]V = 5 x 2^n[/tex]

where n is the number οf days after the videο was pοsted.

Nοw we want tο find οn which day the number οf viewers will be 640. Sο we can set V equal tο 640 and sοlve fοr n:

[tex]640 = 5 x 2^n[/tex]

[tex]2^n = 128[/tex]

n = lοg2(128) = 7

2.   Tο find the first day when mοre than 20,000 peοple will see the videο, we can set V equal tο 20,000 and sοlve fοr n:

[tex]20,000 = 5 x 2^n[/tex]

2^n = 4,000

n = lοg2(4,000) ≈ 11.29

Since n represents the number οf days after the videο was pοsted, we can rοund up tο the next whοle number tο find the first day when mοre than 20,000 peοple will see the videο. Therefοre, οn the 12th day after the videο was pοsted, mοre than 20,000 peοple will see the videο if the trend cοntinues

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Step-by-step explanation:

This is just the corner of a rectangle or a square :

Answer:

28.05$

Step-by-step explanation:

The game got a discount of 15%.
New price is   (30$) ( 0.85 )  =  25.5$   (since the discount is 15% you only pay for the 85% of the original price)

Then an increase of 10%. this is:
New price:   25.5$ (1.10) = 28.05

New price is 28$ with 5 cents

The required probability that a household in Maryland with annual income of ,

$90,000 or more is equal to 0.3377.

$50,000 or less is equal to 0.2218.

Annual household income in Maryland follows a normal distribution ,

Median =  $75,847

Standard deviation = $33,800

Probability of household in Maryland has an annual income of $90,000 or more.

Let X be the random variable representing the annual household income in Maryland.

Then,

find P(X ≥ $90,000).

Standardize the variable X using the formula,

Z = (X - μ) / σ

where μ is the mean (or median, in this case)

And σ is the standard deviation.

Substituting the given values, we get,

Z = (90,000 - 75,847) / 33,800

⇒ Z = 0.4187

Using a standard normal distribution table

greater than 0.4187  as 0.3377.

P(X ≥ $90,000)

= P(Z ≥ 0.4187)

= 0.3377

Probability that a household in Maryland has an annual income of $90,000 or more is 0.3377(rounded to four decimal places).

Probability that a household in Maryland has an annual income of $50,000 or less.

P(X ≤ $50,000).

Standardizing X, we get,

Z = (50,000 - 75,847) / 33,800

⇒ Z = -0.7674

Using a standard normal distribution table

Probability that a standard normal variable is less than -0.7674 as 0.2218. This implies,

P(X ≤ $50,000)

= P(Z ≤ -0.7674)

= 0.2218

Probability that a household in Maryland has an annual income of $50,000 or less is 0.2218.

Therefore, the probability with annual income of $90,000 or more and  $50,000 or less is equal to 0.3377 and 0.2218 respectively.

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To prove that complex numbers with absolute value 1 constitute a group under multiplication, they must satisfy the four axioms of a group which includes

In mathematics, a complex number is described as an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and  can be expressed in the form a + bi, where a and b are real numbers.

The four  axioms of a group includes:

Closure: The product of any two complex numbers with absolute value 1 is another complex number with absolute value 1.

Associativity: The product of any three complex numbers with absolute value 1 is the same irrespective of the order in which the multiplication is performed.

Identity: There exists an element in the group, denoted by 1, such that 1 times any element in the group is equal to that element.

Inverse:  there exists another element in the group, denoted by the reciprocal or inverse, such that their product is equal to the identity element for each element in the group.

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